Determinantal Form for Ladder Operators in a Problem Concerning a Convex Linear Combination of Discrete and Continuous Measures

Abstract

In this contribution we complete and deepen the results in (Huertas et al, Proc Am Math Soc 142(5), 1733–1747, 2014, [1]) by introducing a determinantal form for the ladder operators concerning the infinite sequence $\{Q_{n}(x)\}_{n\ge 0}$ of monic polynomials orthogonal with respect to the following Laguerre–Krall inner product

$$\begin{aligned} \langle f,g\rangle _{\nu } =\int _{0}^{+\infty }f(x)g(x)x^{\alpha } e^{-x}dx+\sum _{j=1}^{m}a_{j}\,f(c_{j})g(c_{j}), \end{aligned}$$

where $c_{j}\in \mathbb {R}_{-} \cup \{0\}$. We obtain for the first time explicit formulas for these ladder (creation and annihilation) operators, and we use them to obtain several algebraic properties satisfied by $Q_{n}(x)$. As an application example, based on the structure of the above inner product $\langle f,g\rangle _{\nu }$, we consider a convex linear combination of continuous and discrete measures that leads to establish an interesting research line concerning the Laguerre–Krall polynomials and several open problems.